Hardy space estimates for limited ranges of Muckenhoupt weights

Abstract

In this article, we give a full necessary and sufficient set of conditions for a Calderón–Zygmund operator to be bounded on weighted Hardy spaces Hwp where w is an Muckenhoupt weight and 0<p<∞. In fact, this result is new even when 1<p<∞ since it allows for Hwp boundedness of an operator when 1<p<q<∞ and w∈Aq, where it is possible that Hwp≠Lwp. These singular integral results are achieved by proving Littlewood–Paley–Stein square function type estimates from Hwp into Lwp for 0<p<∞ and a Muckenhoupt weight w, which are interesting results in their own right. New techniques involving A∞ weight invariant spaces are also used to prove the weighted estimates for Calderón–Zygmund operators. More precisely, we prove the following BMO type weight invariance properties: for a fixed s≥0, the weighted Sobolev-BMO spaces Is(BMOw) coincide for all w∈A∞, the weighted p=∞ type Triebel–Lizorkin spaces F˙∞,ws,2 coincide for all w∈A∞, and these two classes of spaces coincide with each other as well, all of which have comparable norms up to constants depending on an Ap character of the weight w∈A∞.